| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.ClassFieldBridge |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| Enlarging the finite conductor to contain the discriminant primes leaves |
| its infinite part empty and its Artin-kernel property intact. An ordinary |
| positive integer then imposes all of its local congruences. |
| -/ |
|
|
| noncomputable section |
| open scoped NumberField Classical |
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|
| namespace DukePrimeDegree |
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|
| open GlobalClassFieldTheory GlobalClassFieldTheory.GlobalClassFields |
| open PrimeDegreePackets PrimeDegreePackets.GaloisIdealCharacter |
|
|
| variable {K L : Type} [Field K] [NumberField K] [Field L] [NumberField L] |
| variable [Algebra K L] [IsAbelianGalois K L] |
| variable [NumberField.IsTotallyReal L] |
|
|
| theorem exists_even_unramified_artin_modulus : |
| ∃ m : RayClass.Modulus K, m.infinitePart = ∅ ∧ |
| m.congruenceSubgroup ≤ (ideleClassNorm K L).range ∧ |
| ∀ v : IsDedekindDomain.HeightOneSpectrum (𝓞 K), v ∉ m.finitePart.support → |
| Algebra.IsUnramifiedAt (𝓞 K) (primeAbove K L v).val := by |
| obtain ⟨m, hm, hnorm⟩ := exists_even_artin_modulus K L |
| let D : 𝓞 K := (NumberField.discr L).natAbs |
| have hD : D ≠ 0 := by |
| change ((NumberField.discr L).natAbs : 𝓞 K) ≠ 0 |
| exact_mod_cast (Int.natAbs_pos.mpr (NumberField.discr_ne_zero L)).ne' |
| let I := integralIdealFractionalUnit (Ideal.span {D}) |
| (by simpa only [ne_eq, Ideal.span_singleton_eq_bot] using hD) |
| let b : RayClass.FiniteModulus K := |
| (FractionalIdealGroup.countVector I).mapRange Int.natAbs rfl |
| let m' := m.replaceFinitePart (m.finitePart + b) |
| have hle : m ≤ m' := by |
| constructor |
| · exact le_add_of_nonneg_right (by positivity) |
| · exact Finset.Subset.refl _ |
| refine ⟨m', hm, (RayClass.Modulus.congruenceSubgroup_antitone hle).trans hnorm, ?_⟩ |
| intro v hv |
| have hv0 : m'.finitePart v = 0 := Finsupp.notMem_support_iff.mp hv |
| have hb : b v = 0 := by |
| change m.finitePart v + b v = 0 at hv0 |
| omega |
| have hcount : FractionalIdeal.count K v (Ideal.span {D}) = 0 := by |
| have hc : Int.natAbs (FractionalIdeal.count K v (Ideal.span {D})) = 0 := hb |
| exact Int.natAbs_eq_zero.mp hc |
| have hnot : D ∉ v.asIdeal := by |
| exact mt ((principal_count_ne_zero_iff_mem K D hD v).mpr) (by simpa using hcount) |
| let : (primeAbove K L v).val.IsPrime := (primeAbove K L v).property.1 |
| exact unramified_of_absDiscr_not_mem K L v.asIdeal (primeAbove K L v).val |
| (primeAbove K L v).property.2.over.symm hnot |
|
|
| theorem exists_integer_even_artin_modulus : |
| ∃ (m : RayClass.Modulus K) (N : ℕ), 0 < N ∧ |
| (NumberField.discr L).natAbs ∣ N ∧ m.infinitePart = ∅ ∧ |
| m.congruenceSubgroup ≤ (ideleClassNorm K L).range ∧ |
| (∀ v ∈ m.finitePart.support, |
| algebraMap (𝓞 K) (v.adicCompletionIntegers K) (N : 𝓞 K) ∈ |
| IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) ^ m.finitePart v) ∧ |
| ∀ v : IsDedekindDomain.HeightOneSpectrum (𝓞 K), v ∉ m.finitePart.support → |
| Algebra.IsUnramifiedAt (𝓞 K) (primeAbove K L v).val := by |
| obtain ⟨m, hm, hnorm, hur⟩ := exists_even_unramified_artin_modulus (K := K) (L := L) |
| obtain ⟨N, hN, hdN, hlocal⟩ := exists_integer_for_local_congruences |
| m.finitePart.support m.finitePart (Int.natAbs_pos.mpr (NumberField.discr_ne_zero L)) |
| exact ⟨m, N, hN, hdN, hm, hnorm, hlocal, hur⟩ |
|
|
| end DukePrimeDegree |
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| end |
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| end OAI |
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|
